The Complicated Phase Space of a 2D Nonlinear Dynamical System: Homoclinic Tangles and Heteroclinic Intersections (pp. 211-227)
Authors: Christos Tsironis
Abstract: The intricate phase-space transport in a dynamical system characterized by incomplete chaos, even for large perturbations, is studied in terms of the geometry of an unstable periodic orbit with multiplicity 3 on a surface of section representing a 2D Hamiltonian system. The homoclinic tangle of the orbit consists of two resonance areas, in contrast with the tangles considered in previous studies that involve only one resonance area. The properties of the homoclinic tangle are presented and analyzed, and also considerations regarding the onset of heteroclinic intersections are made, based on numerical results in a wide interval of energies. It is verified that the tracking of the intersections of the inner and outer lobes belonging to the same resonance, as well as to both resonance areas, concludes in certain rules which allow the derivation of quantitative relations about the number of intersections and the understanding of the complex behavior of the higher-order lobes in terms of lower-order lobes.